In
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
, Hilbert's irreducibility theorem, conceived by
David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and philosopher of mathematics and one of the most influential mathematicians of his time.
Hilbert discovered and developed a broad range of fundamental idea ...
in 1892, states that every finite set of
irreducible polynomial
In mathematics, an irreducible polynomial is, roughly speaking, a polynomial that cannot be factored into the product of two non-constant polynomials. The property of irreducibility depends on the nature of the coefficients that are accepted f ...
s in a finite number of variables and having
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.
Formulation of the theorem
Hilbert's irreducibility theorem. Let
:
be irreducible polynomials in the ring
:
Then there exists an ''r''-tuple of rational numbers (''a''
1, ..., ''a
r'') such that
:
are irreducible in the ring
:
Remarks.
* It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible specializations, called Hilbert set, is large in many senses. For example, this set is
Zariski dense in
* There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (''a''
1, ..., ''a
r'') to be integers.
* There are many
Hilbertian fields, i.e., fields satisfying Hilbert's irreducibility theorem. For example,
number fields are Hilbertian.
[Lang (1997) p.41]
* The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take
in the definition. A result of Bary-Soroker shows that for a field ''K'' to be Hilbertian it suffices to consider the case of
and
absolutely irreducible, that is, irreducible in the ring ''K''
alg 'X'',''Y'' where ''K''
alg is the algebraic closure of ''K''.
Applications
Hilbert's irreducibility theorem has numerous applications in
number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
. For example:
* The
inverse Galois problem, Hilbert's original motivation. The theorem almost immediately implies that if a finite group ''G'' can be realized as the Galois group of a Galois extension ''N'' of
::
:then it can be specialized to a Galois extension ''N''
0 of the rational numbers with ''G'' as its Galois group.
[Lang (1997) p.42] (To see this, choose a monic irreducible polynomial ''f''(''X''
1, ..., ''X
n'', ''Y'') whose root generates ''N'' over ''E''. If ''f''(''a''
1, ..., ''a
n'', ''Y'') is irreducible for some ''a
i'', then a root of it will generate the asserted ''N''
0.)
* Construction of elliptic curves with large rank.
[
* Hilbert's irreducibility theorem is used as a step in the ]Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Ferma ...
proof of Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
.
* If a polynomial